Single-Particle Electrodynamics - Assassination Science

Now, the torque on an object (around its centre of energy) is simply the
time rate of change of its mechanical spin angular momentum,
N ≡ ṡ,
in the same way that the force is the time rate of change of its linear mechanical
momentum,
F ≡ ṗ.
In general, the spin angular momentum s of a classical object may be of any
value, and point in any direction. In particular, there is no good reason why
it should point in the same direction as the electric dipole moment: it can
do so (e.g., if one uses the rigid stick above as an axis for a rotating flywheel,
and one assumes the charges and the rigid stick to be themselves massless);
but it need not, in general.
On the other hand, we shall often wish to apply the equations of motion
derived in this thesis to the case of a spin-half particle, such as an electron.
For such a particle, we know that the mass must be a constant of the motion
(namely, m e ); and we know that the magnitude s of its spin angular momentum
must be a constant of the motion (namely, 2¯h). But we also know that
1
any dipole moment of a spin-half particle must be parallel to its spin vector
s (basically because there are no other three-vectors available in the Dirac
algebra):
d ≡ dσ,
s ≡ sσ ≡ 2¯hσ. 1
With such a parallelism identity in effect, we may use the torque result N
to obtain the precession rate very simply:
d˙ ∣ = d
d‖s sṡ ≡ d N. (4.12)
s
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